Almost every set in exponential time is P-bi-immune
A set A is P-bi-immune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of P-bi-immune languages in linear-exponential time (E). Using resource-bounded measure, we prove that the class of P-bi-immune sets has measure 1 in E. This implies that “almost” every language in E is P-bi-immune. A bit further, we show that every p-random (pseudorandom) language is E-bi-immune. Regarding the existence of P-bi-immune sets in NP, we show that if NP does not have measure 0 in E, then NP contains a P-bi-immune set. Another consequence is that the class of ≤ 1-tt P -complete languages for E has measure 0 in E. In contrast, using the approach of resource-bounded category, it is shown that in E, the class of P-bi-immune languages lacks the property of Baire (the Baire category analogue of Lebesgue measurability).
KeywordsComplexity Class SIAM Journal Random Oracle Exponential Time Meaningful Measure
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